Exemplu De Vibratie A Unei Corzi.docx

In vibrational analysis, modes of vibration are the different types in which the system tries to oscillate naturally, i.e. without any excitation force. The frequency of oscillation is termed as modal frequency (or natural frequency) and the shape made by the system is called mode shape. Now, there can be various types in which a system can vibrate but the system has a tendency to follow certain shapes (and an associated frequency) while oscillating depending upon its 1. Material property (Density, Young's modulus, Damping etc.) 2. Governing mechanics (Governing equation obtained after applying Newton's (Force balance) or Energy approach) 3. Boundary conditions (The way in which the system is held at the boundaries). Physically it can be seen with the help of an example of a vibrating string. When we define the mechanics of the string using Newton's approach, we get a second order partial differential equation in space and time, solution of which is sinusoidal in both space and time.

Now, if the string is fixed at the two ends, when we apply the boundary conditions,

The solution is a sine wave in space. Thus, it has tendency to vibrate in the following exact shapes as shown in the figure.

Notice that if we change any of the parameters mentioned above, the modes will change 1. Material property (Changing the density or Young's modulus will change the frequency) 2. Governing mechanics (The above equation is derived assuming small oscillations. For large oscillations, the PDE is not linear and the solution is not sinusoidal). 3. Boundary conditions (if we change the boundary conditions to let's say fixed free, while the nature of solution still remains same, the frequency and the shape of the mode shapes changes. In fact, it becomes a cosine wave in this case). Point No. 1 is employed in a lot of musical instruments, seen clearly in a guitar where strings of different thicknesses and creation of local nodes by changing finger position is used to change the nature of emitted sound. The reason why study of modes of vibration is really important in vibrational analysis is that it is found that any vibration response can be written as a sum of individual mode shapes because of their nice properties such as orthogonality and completeness. (Modal superposition method) Finally, of all the modes, which ones will be dominant depends on the initial conditions, i.e. initial shape of the string in case of free response and excitation frequency and spatial variation of forcing function in forced response (Modal participation factor). This is not so straightforward to see even for simple systems but a general rule of thumb is that the contribution of lower modes is higher (intuitively because the number of nodes (points of zero displacement) is less and hence it is easy to excite the system in that shape). This is true unless the forcing function is pretty close (or equal) to one of the modal frequencies during which the system tends to vibrate in that frequency. (Resonance)